Metric Properties of Conditional Moments with Exponential Weights for Model Adequacy

Abstract

The paper proposes a new integrated conditional moment test for model adequacy related to the tests studied in Bierens (1982) and Bierens and Ploberger (1997). The new test allows for a numerical calculation of its asymptotic distribution when the parameter estimator is asymptotically linear. We find that the power of the test in misspecified linear models is better or similar to some of the most commonly used alternatives in the literature. The metric properties of the proposed test are used to study the impact of three types of aggregation on the specification error - aggregation of observations across time, cross-sectional aggregation of variables, or aggregation of different models for the same variable. We show that neglected non-linearity in linear models is asymptotically negligible with a power-type rate of decay in the case of independent observations when data are aggregated across time. We provide an illustration from the field of finance with the capital asset pricing model (CAPM). The frequency of rejection of the linear specification of the CAPM can decline more than three times when the model is estimated with monthly rather than daily returns.